Membrane Theory

We examine the individual terms of the force equilibrium equations, (3.17a-c), and the moment equilibrium equations, (3.22a-c). We see that the equations are coupled only through the transverse shear stress resultants, Q _α and Q _β If we suppose that for a certain class of shells, the stress couples are an order of magnitude smaller than the extensional and in-place shear stress resultants, we may deduce from equations (3.22a-c) that the transverse shear stress resultants are similarly small and thus may be neglected in the force equilibrium equations, (3.17). This implies that the shell may achieve force equilibrium through the action of in-plane forces alone. From a physical viewpoint, this possibility is evident for the first two equilibrium equations which reflect in-plane resistance to in-plane loading, a natural and obvious mechanism. On the other hand, the third equilibrium equation refers to the normal direction, and the possibility of resisting transverse loading with in-plane forces alone is not as apparent. It is evident from equation (3.17c) that this mode of resistance is possible only if at least one radius of curvature is finite; i.e., R _α and/or R _β ≠∞. Thus, flat plates are excluded from resisting transverse loading in this manner, within the limitations of small deformation theory (assumption [2], table 1-1.